The knot quandle of the twist-spun trefoil is a central extension of a Schl\"{a}fli quandle

TL;DR

This paper introduces Schl"afli quandles related to tessellations and demonstrates that the knot quandle of the $m$-twist-spun trefoil is a central extension of such a quandle, linking knot theory and geometric symmetries.

Contribution

It defines Schl"afli quandles and establishes that the knot quandle of the $m$-twist-spun trefoil is a central extension of these quandles, connecting knot invariants with tessellation symmetries.

Findings

Knot quandle of the $m$-twist-spun trefoil is a central extension of a Schl"afli quandle.

Introduction of Schl"afli quandles related to regular tessellations.

Establishes a new link between knot theory and geometric tessellations.

Abstract

A quandle is an algebraic system which excels at describing limited symmetries of a space. We introduce the concept of Schl\"{a}fli quandles which are defined relating to chosen rotational symmetries of regular tessellations. On the other hand, quandles have a good chemistry with knot theory. Associated with a knot we have its knot quandle. We show that the knot quandle of the $m$-twist-spun trefoil is a central extension of the Schl\"{a}fli quandle related to the regular tessellation $\{ 3, m \}$ in the sense of the Schl\"{a}fli symbol if $m \geq 3$.